The \(\sigma \) -algebra generated by the intervals \(\left\lbrace  I_{n}\left(x\right) :x\in G_{m}\right\rbrace  \)  will be denoted by \(\digamma _{n}\)  \(\left(n\in \mathbb {N}\right) .\)  Denote by \(f=\left( f^{\left( n\right) },n\in \mathbb {N}\right) \)  a martingale with respect to \(\digamma _{n}\)  \(\left(n\in \mathbb {N}\right)\)  (for details see e.g. [1]}, [2]}, [3]}). The maximal
function of a martingale \(f\)  is defend by       
\(f^{\ast }=\sup _{n\in \mathbb {N}}\left|f^{\left( n\right) }\right|.\) 
